Bring broth and next 2 ingredients to a boil in a large saucepan over medium-high heat.
Whisk in grits, and return to a boil; reduce heat to low, and stir in butter. Cover and simmer, stirring occasionally, 10 minutes or until liquid is absorbed.
Remove from heat.
Stir together eggs and next 4 ingredients in a large bowl. Gradually stir about one-fourth of hot grits mixture into egg mixture; add egg mixture to remaining hot grits mixture, stirring constantly. Stir in shrimp until blended.
Pour grits mixture into a lightly greased 11- x 7-inch baking dish.
Bake at 325 for 55 minutes to 1 hour or until mixture is set.